A product of logarithms

$\begin{aligned} P & = \log_2 3 \times \log_3 4 \times \log_4 5 \times \cdots \log_{1023} 1024 & \small \color{#3D99F6} \text{Using }\log_b a = \frac {\log a}{\log b} \\ & = \frac {\cancel{\log 3}}{\log 2} \times \frac {\cancel{\log 4}}{\cancel{\log 3}} \times \frac {\cancel{\log 5}}{\cancel{\log 4}} \times \cdots \times \frac {\log 1024}{\cancel{\log 1023}} \\ & = \frac {\log 1024}{\log 2} = \frac {\log 2^{10}}{\log 2} = \frac {10\cancel{\log 2}}{\cancel{\log 2}} = \boxed{10} \end{aligned}$

Since $\log_y(x) = \frac{\log_a(x)}{\log_a(y)}$ , this is equal to

$\frac{\log_2(3)}{\log_2(2)}\cdot\frac{\log_2(4)}{\log_2(3)}\cdot\cdot\cdot \frac{\log_2(1024)}{\log_2(1023)} = \frac{\log_2(1024)}{\log_2(2)} = 10$